I'll be up front and say I have a very strong bias against organizations such as that. If Dante's version of the afterlife is correct, there is a very special place for those people when they die.
That said, there's a grain of truth in what they wrote.
But first, Alexander Fleming realized the true risks of using antibiotics in his 1945 Nobel Prize acceptance speech:
But I would like to sound one note of warning. Penicillin is to all intents and purposes non-poisonous so there is no need to worry about giving an overdose and poisoning the patient. There may be a danger, though, in underdosage. It is not difficult to make microbes resistant to penicillin in the laboratory by exposing them to concentrations not sufficient to kill them, and the same thing has occasionally happened in the body.
I'll look at the first claim later. The second claim is becoming reality. There are a number of extremely dangerous bacteria that are now immune to most, if not all, antibiotics. The World Health Organization, along with many other organizations, is worried that Fleming's concern has become a severe problem. We may well be entering the post-antibiotic age. In the end, everything we do to defeat the immense powers of evolution are best short-lived solutions.
The problem is that we haven't discovered any new classes of antibiotics for thirty years. Thirty calendar years in terms of a dog's life is a long time. In terms of a bacteria's life, it's a long, long time. We may well be entering an age where a paperclip cut can kill. Those "natural" sites do not recognize that. Paperclip cuts could kill are 70 years ago, but not now. Those days will appear again.
Let's look at that first claim. While antibiotics are generally much more effective against gram positive than gram negative bacteria, using antibiotics on a massive scale kills almost kill all bacteria. Our gut bacteria are mostly gram negative.
Recent discoveries show that our immune system apparently depends,at least in part, on the presence of a healthy gut bacteria. Fleming's statement that "penicillin is to all intents and purposes non-poisonous so there is no need to worry about giving an overdose and poisoning the patient" is not correct.
However, it's important to remember Isaac Asimov's distinction between "wrong" and "not even wrong". The naturalists who reject antibiotics (and immunizations, which the linked articles reject) are "not even wrong." They are, IMNHO, worse than not even wrong. They are evil.